ee541F11HWSolutions05-06

ee541F11HWSolutions05-06 - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #05-06 73 Fall Semester, 2011 U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: Solutions, Homework #05-#06 Fall, 2011 Due: 10/25/2011 Choma Solutions Problem #25: The active filter shown in Figure (P25a) utilizes three ideal operational transconductor am- plifiers and a linear network whose transfer function, V x /V i , is H(s) . Observe that two of the transconductors have identical I/O transconductances of G m , while the third transconductor am- plifier has an I/O transconductance of 2G m . Derive a general expression for the input -to- output (I/O) voltage gain, A v (s) = V o /V i . G m G m 2G m H(s) V o V i V x Figure (P25) Figure (P25.1) redraws the circuit schematic diagram of Figure (P25) for the express purpose of delineating the pertinent branch currents that flow in the given network, as well as the voltages es- tablished at relevant nodes. KCL at the output node of the system yields G m G m 2G m H(s) V o V i V 0 V i V x 2G V mx GV mi mo 0 0 0 0 0 0 0 0 Figure (P25.1) 2G V G V G V ,  (P25-1) which breeds x io 2V V V . (P25-2) Since V x is the output voltage of the system block characterized by the transfer function, H(s) , x i V H(s)V . (P25-3) Upon inserting this result into (P25-2), we arrive at
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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #05-06 74 Fall Semester, 2011 o v i V A(s ) 2H(s ) 1. V  (P25-4) Solutions Problem #26: The network of Figure (P25) is exploited by realizing the transfer function, H(s) , as the subcircuit comprised of a capacitance, C , and a transconductor whose transconductance is G mi , as is depicted in Figure (P26). G m G mi V o V i G m 2G m V x C Figure (P26) (a). Determine the I/O voltage transfer function, A v (s) = V o /V i . A comparison of the filter network in Figure (P26) indicates that it is topologically identical to the generic form of this filter, which is depicted in Figure (P25). The only difference between the two diagrams is that instead of symbolically representing node voltage V x as the product of input vol- tage V i and transfer function H(s) , voltage V x is depicted herewith as being generated by a subcir- cuit comprised of a transconductor of forward transconductance G mi and a capacitance, C , which returns the output port of this transconductor amplifier to ground. We therefore need only to derive the expression for the transfer function, V x /V i , in terms of G mi and C and then substitute this result into (P25-4) to establish an analytical relationship for the network voltage gain, A v (s) = V o /V i . To this end, we have extracted the G mi C subcircuit as Figure (P26.1) and have shown relevant node voltages and branch currents on this subcircuit. From the figure before us, G mi V i V x V x C G(V V ) mi x i V ) mi x i 0 Figure (P26.1)  x mi x i 1 VG V V , sC  (P26-1) which readily engenders mi xm i mi i mi sC G sC H(s) .
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ee541F11HWSolutions05-06 - EE 541 University of Southern...

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